The value of  nC1+n+1C2+n+2C3+⋯+n+m−1Cm is equal to

nC1+n+1C2+n+2C3+…+n+m−1Cm =nCn−1+n+1Cn−1+n+2Cn−1+…+n+m−1Cn−1 = Coefficient of xn−1 in (1+x)n+(1+x)n+1+(1+x)n+2+…+(1+x)n+m−1 = Coefficient of xn−1 in (1+x)n(1+x)m−1−1(1+x)−1 = Coefficient of xn−1 in (1+x)m+n−(1+x)nx = Coefficient of xn in (1+x)m+n−(1+x)n =m+nCn-1Similarly, we can prove mC1+m+1C2+m+2C3+…+m+n−1Cn=m+nCm-1